Analytic functions in complex-variable function occupies very important position, it has important properties and widely used. Analytic functions throughout the equivalence conditions in complex-variable function in teaching material of the always, several appear, therefore, grasps the definition and analytic functions when the contact is very important. Because the definition of analytic functions in different writings form different, firstly, this paper defines the analytic functions of eight different definitions, by using cauchyschwartz integral theorem, Taylor theorem and some properties of binary function, the method of the push proves this in turn eight different defined conditions equivalent to each other by proof make readers. The theory of analytic functions had a deeper understanding, but also for future application analytic functions to lay a solid foundation.%解析函数是复变函数论研究的主要对象,它具有重要的性质和广泛应用.本文在教材内容的基础上,进一步探讨了解析函数的几个等价条件及各等价条件的证明,文章首先给出了解析函数的五种不同的定义,通过运用柯西积分定理、泰勒定理、导数的以及顺推的方法,依次证明了这五种不同定义彼此相互等价；加深了对解析函数的理论的理解,为以后应用解析函数打下了坚实的基础.
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